7.1 The restricted Weyl group and the maximal split “subalgebra”

Let be any real form of the complex Lie algebra , its Cartan involution, and let

be the corresponding Cartan decomposition. Furthermore, let

be a maximal noncompact Cartan subalgebra, with (respectively, ) its compact (respectively,
noncompact) part. The real rank of is, as we have seen, the dimension of . Let now denote the
root system of , the restricted root system and the multiplicity of the restricted root
.

As explained in Section 4.9.2, the restricted root system of the real form can be either reduced or
non-reduced. If it is reduced, it corresponds to one of the root systems of the finite-dimensional simple Lie
algebras. On the other hand, if the restricted root system is non-reduced, it is necessarily of
-type [93] (see Figure 19 for a graphical presentation of the root system).

The restricted Weyl group

By definition, the restricted Weyl group is the Coxeter group generated by the fundamental reflections,
Equation (4.55), with respect to the simple roots of the restricted root system. The restricted Weyl group
preserves multiplicities [93].

The maximal split “subalgebra”

Although multiplicities are an essential ingredient for describing the full symmetry , they turn out to be
irrelevant for the construction of the gravitational billiard. For this reason, it is useful to consider the
maximal split “subalgebra”, which is defined as the real, semi-simple, split Lie algebra with the
same root system as the restricted root system as (in the -case, we choose for
definiteness the root system of to be of -type). The real rank of coincides with the rank
of its complexification , and one can find a Cartan subalgebra of , consisting of
all generators of which are diagonalizable over the reals. This subalgebra has the
same dimension as the maximal noncompact subalgebra of the Cartan subalgebra of
.

By construction, the root space decomposition of with respect to provides the same root system
as the restricted root space decomposition of with respect to , except for multiplicities, which are
all trivial (i.e., equal to one) for . In the -case, there is also the possibility that
twice a root of might be a root of . It is only when is itself split that and
coincide.

One calls the “split symmetry algebra”. It contains as we shall see all the information about the
billiard region [95]. How can be embedded as a subalgebra of is not a question that shall be of our
concern here.

The Iwasawa decomposition and scalar coset Lagrangians

The purpose of this section is to use the Iwasawa decomposition for described in Section 6.4.5 to
derive the scalar Lagrangian based on the coset space . The important point is to understand
the origin of the similarities between the two Lagrangians in Equation (5.45) and Equation (7.8)
below.

The full algebra is subject to the root space decomposition

with respect to the restricted root system. For each restricted root , the space has dimension .
The nilpotent algebra , consisting of positive root generators only, is the direct sum

over positive roots. The Iwasawa decomposition of the U-duality algebra reads

(see Section 6.4.5). It is that appears in Equation (7.5) and not the full Cartan subalgebra since
the compact part of belongs to .

This implies that when constructing a Lagrangian based on the coset space , the only part
of that will show up in the Borel gauge is the Borel subalgebra

Thus, there will be a number of dilatons equal to the dimension of , i.e., equal to the real rank of ,
and axion fields for the restricted roots (with multiplicities).

More specifically, an (-dependent) element of the coset space takes the form

where the dilatons and the axions are coordinates on the coset space, and where denotes an
arbitrary set of parameters on which the coset element might depend. The corresponding Lagrangian
becomes

where the sums over are sums over the multiplicities of the positive restricted roots
.

By comparing Equation (7.8) with the corresponding expression (5.45) for the split case, it is
clear why it is the maximal split subalgebra of the U-duality algebra that is relevant for the
gravitational billiard. Were it not for the additional sum over multiplicities, Equation (7.8) would
exactly be the Lagrangian for the coset space , where is the maximal
compact subalgebra of (note that ). Recall now that from the point of view of the
billiard, the positive roots correspond to walls that deflect the particle motion in the Cartan
subalgebra. Therefore, multiplicities of roots are irrelevant since these will only result in several walls
stacked on top of each other without affecting the dynamics. (In the -case, the wall
associated with is furthermore subdominant with respect to the wall associated with when
both and are restricted roots, so one can keep only the wall associated with .
This follows from the fact that in the -case the root system of is taken to be of
-type.)